This paper investigates a peculiar case of thermal convection in a vertical porous prism with impermeable and partially conducting walls. We facilitate the analysis in the numerical finite-element environment alongside with analytical considerations, in special cases where direct solutions are feasible. The present eigenvalue problem results in a non-normal-mode behaviour in the horizontal cross-sectional plane. Further, it is identified that the stagnation points for the horizontal flow are displaced from the extremal points of the temperature perturbation, for both symmetric and antisymmetric eigenfunctions. In addition, the corresponding normal-mode counterparts are provided from an analogy solution. We show that the critical Rayleigh number decreases with increasing Robin parameter values for all of the investigated aspect ratios. Finally, the influence of the aspect ratio on the critical Rayleigh number for the fully conducting wall case is identified. An asymptotic benchmark case of the Robin condition is validated from well-known analytical solutions which confirm the effectiveness of the predictions made in this paper. In fact, this is the first contribution that reports a three-dimensional geometry with a two-dimensional non-normal mode.

中文翻译：

---

具有罗宾型热壁条件的三角形多孔棱镜中的对流开始

本文研究了具有不渗透和部分传导壁的垂直多孔棱柱中的热对流的特殊情况。在直接解决方案可行的特殊情况下，我们促进了数值有限元环境中的分析以及分析考虑。当前的特征值问题导致水平横截面中的非正常模式行为。此外，对于对称和反对称本征函数，水平流的驻点都偏离了温度扰动的极值点。此外，对应的正常模式对应物是从类比解决方案中提供的。我们表明，对于所有研究的纵横比，临界瑞利数随着罗宾参数值的增加而降低。最后，确定了纵横比对全导电壁情况的临界瑞利数的影响。Robin 条件的渐近基准案例从众所周知的分析解决方案中得到验证，这些解决方案证实了本文所做预测的有效性。事实上，这是第一个报告具有二维非正常模式的三维几何的贡献。